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Time Fractional Formalism: Classical and Quantum Phenomena

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Time Fractional Formalism: Classical and Quantum Phenomena Time Fractional Formalism: Classical and Quantum Phenomena * Hosein Nasrolahpour Abstract In this review, we present some fundamental classical and quantum phenomena in view of time fractional formalism. Time fractional formalism is a very useful tool in describing systems with memory and delay. We hope that this study can provide a deeper understanding of the physical interpretations of fractional derivative. Keywords: Fractional calculus; Fractional classical mechanics; Fractional classical electromagnetism; Fractional quantum mechanics. This work is dedicated to the soul of my father 1- Introduction Fractional calculus is a very useful tool in describing the evolution of systems with memory, which typically are dissipative and to complex systems. Complex systems include very broad and general class of systems and materials. For instance, glasses, biopolymers, biological cells, porous materials, amorphous semiconductors and liquid crystals can be considered as complex systems. Scaling laws and self-similar behavior are supposed to be fundamental features of complex systems. In recent decades the fractional calculus and in particular the fractional differential equations has attracted interest of researches in several areas including mathematics, physics, chemistry, biology, engineering and economics [1-4].Applications of fractional calculus in the field of physics have gained considerable popularity and many important results were obtained during the last years. Some of the areas of these applications include: classical mechanics [8-11], classical electromagnetism [32-38], special relativity [39, 40], non-relativistic quantum mechanics [43-50] and relativistic quantum mechanics and field theory [51-58]. Despite these various applications, there are some important challenges. For example physical interpretation for the fractional derivative is not completely clarified yet. In this review, we aim to present some aspects of physical interpretation for the fractional derivative by studying the behavior of fundamental classical and quantum phenomena within the framework of time fractional formalism. In the following, fractional calculus is briefly reviewed in Sec. 2. The fractional relaxation and oscillation process are discussed in Sec. 3. Time fractional Maxwell’s equations are presented in sec. 4. In Sec. 5 time fractional Schrödinger equation and time fractional Pauli equation are given. Finally in Sec. 6, we will present our summary and discussion. 2- Mathematical tools: Fractional calculus Although the application of Fractional calculus has attracted interest of researches in recent decades, it has a long history when the derivative of order 0.5 has been described by Leibniz in a letter to L'Hospital in 1695. Fractional calculus is the calculus of derivatives and integrals with arbitrary (real or even complex) order, which unify and generalize the notions of integer order differentiation and n-fold integration, which have found many applications in recent studies to model a variety processes from classical to quantum physics. In the following, we briefly revisit essentials of fractional calculus. 2.1. The Caputo fractional derivative operator The commonest way to obtain a fractional differential equation for describing the evolution of a typical system is to generalize the ordinary derivative in the standard differential equation into the fractional derivative. Fractional differential equation can be include for instance derivative of order 0.5, 2 , and *Correspondence: Hosein Nasrolahpour, E-mail: [email protected] so on. Since the age of Leibniz various types of fractional derivatives have been proposed. In fact, the definition of the fractional order derivative is not unique and there exists several definitions including, Grünwald–Letnikov, Riemann-Liouville, Weyl, Riesz and Caputo for fractional order derivative. Fractional differential equations defined in terms of Caputo derivatives require standard boundary (initial) conditions. Also the Caputo fractional derivative satisfies the relevant property of being zero when applied to a constant. For these reasons, in this paper we prefer to use the Caputo fractional derivative. The left (forward) Caputo fractional derivative of a time dependent function f() t is defined by 1 t (1) cD f()()() t t n 1 f ( n ) d 0,t 0 0 t ()n 0 Where, n is an integer number and is the order of the derivative such that n-1< <n and f ()n () denotes the n-th derivative of the function f () . For example when is between 0 and 1, we have 1t f ( ) c D f()() t t d 0 1 (2) 0 t (1 ) 0 As we can see from the above equations Caputo derivative implies a memory effects by means of a convolution between the integer order derivative and a power of time. Also the Laplace transform to Caputo's fractional derivative gives n 1 c m 1 ( m ) (3) L{0 Dt f ( t )} s F ( s ) s f (0) m 0 where, F() s is the Laplace transform of f() t . 2.2. The Mittag-Leffler function During the recent years the Mittag-Leffler (ML) function has caused extensive interest among physicist due to its role played in describing realistic physical systems with memory and delay. It was originally introduced by G.M. Mittag-Leffler in 1902[5]. The ML function is such a one-parameter function defined by the series expansion as z k (4) E () z ∈ , 0 k 0 (1 k ) And its general two-parameter representations is defined as k z (5) E, () z ∈ , ∈ , 0 k 0 () k where is the set of complex numbers and ( ) denotes the Gamma function. This function is in fact a generalization of the exponential function. For example, for the special case of 1, the ML function z Eq. (4) reduces to the exponential function E1() z e . Furthermore, since the ML function generalizes the exponential function, the Euler identity for an exponential function with a complex argument (i.e., ei cos( ) i sin( ) ) can also be written for the ML function in a similar manner. So we have (6) E( i ) cos ( ) i sin ( ) Where sin ( ) and cos ( ) are sine and cosine ML functions respectively and defined as ( 1)n ( )2 n 1 ( 1)n ( )2 n sin ( ) , cos ( ) (7) n 0 ((2n 1) 1) n 0 (2n 1) Also, it is notable that although exponential function possesses the semigroup property (i.e., a() z1 z 2 az 1 az 2 e e e ) the function E () az does not possess the semigroup property in general [6] (this property leads to important results in fractional quantum mechanics [48]). Mittag-Leffler function, as a generalized exponential function, naturally arises in the solutions of ordinary differential equations of arbitrary (non-integer) order. Therefore the Laplace transform for ML function will be very useful in solving fractional differential equations: m! s L{ tm 1 E ( m ) ( t )} (8) , ()s m 1 1 Wheres . 3- Classical mechanics: fractional relaxation and fractional oscillation The fundamental processes in physics are described by equations for the time evolution of a quantity X() t in the form: dX() t (9) LX() t dt where L can be both linear or nonlinear operator. For instance there are many relaxation phenomena in nature whose relaxation function obeys the simple approximate equation dx() t (10) x( t ) 0 dt We can write the above equation as dx( t ) 1 x() t (11) dt The solution of the above equation is the normalized exponential Debye-relaxation function (i.e. t x() t e ), with relaxation time . However, there are some experimental evidences that relaxation in several complex disordered systems deviates from the classical exponential Debye pattern [12-24]. Nowadays, it has proved that the fractional relaxation equation can be a successful mathematical construct that reflects the main features of evolution of such systems. The commonest way to obtain a fractional differential equation for describing the evolution of a typical system is to generalize the ordinary derivative in the standard differential equation into the fractional derivative d1 d (12) dt1 dt d where denotes the Caputo’s derivative operator of order and, is a new parameter representing dt the fractional time components in the system[32] and its dimension is the second. In the case 1the expression transforms into ordinary time derivative operator 1 d d (13) 1dt 1 dt Therefore we can easily arrive at the fractional relaxation equation by changing the first order derivative in the Eq. (10) to a derivative of an arbitrary order: d x() t x( t ) 0 0 1 (14) 1dt with the solution: t (15) x( t ) x (0) E ( 1 ( )) It is showed that this solution and this model for the relaxation processes can be successfully adopted to interpret experimental data on relaxation in several complex disordered systems. The second example is the simple harmonic oscillator .The harmonic oscillator, given by the well-known second order linear differential equation with constant coefficients d2 x (16) m kx 0 dt 2 is a cornerstone of classical mechanics [7]. We can obtain the differential equation of a simple fractional oscillator [25-31] by changing the second derivative in the harmonic oscillator equation to a derivative of an arbitrary order (Eq. (12)): 2 m d x (17) kx 0 0 1 2(1 )dt 2 We can write the above equation also as 2 d x2(1 ) k x 0 (18) dt2 m where the parameter f defined by 2 2(1 ) 2 (19) f k and 2 , so we can rewrite the fractional differential equation of the system as m d2 x 2x 0 (20) dt 2 f The solution of the above equation reads: 22 2 2 (21) x() t x (0)( E2 f t )(0) x tE 2 ,2 ( f t ) Now if we choose x (0) 1and x(0) 0 as the initial condition, the solution becomes x()() t E 2 t 2 (22) 2 f We can easily see that as 1, above equations gives (23) 2 2 2 2 2 2 E2 (f t ) E 2 ( t ) cosh( t ) cosh( i t ) cos( t ) As we can see from Eq.
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